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agentica-org/DeepScaleR-Preview-Dataset | A state requires that all boat licenses consist of the letter A or M followed by any five digits. What is the number of groups of letters and numbers available for boat licenses? | 200000 |
|
agentica-org/DeepScaleR-Preview-Dataset | In some cells of a \(10 \times 10\) board, there are fleas. Every minute, the fleas jump simultaneously to an adjacent cell (along the sides). Each flea jumps strictly in one of the four directions parallel to the sides of the board, maintaining its direction as long as possible; otherwise, it changes to the opposite direction. Dog Barbos observed the fleas for an hour and never saw two of them on the same cell. What is the maximum number of fleas that could be jumping on the board? | 40 |
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agentica-org/DeepScaleR-Preview-Dataset | A total of 17 teams play in a single-elimination tournament. (A single-elimination tournament is one where once a team has lost, it is removed from the competition.) How many total games must be played before a winner can be declared, assuming there is no possibility of ties? | 16 |
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agentica-org/DeepScaleR-Preview-Dataset | A zealous geologist is sponsoring a contest in which entrants have to guess the age of a shiny rock. He offers these clues: the age of the rock is formed from the six digits 2, 2, 2, 3, 7, and 9, and the rock's age begins with an odd digit.
How many possibilities are there for the rock's age? | 60 |
|
agentica-org/DeepScaleR-Preview-Dataset | Perpendiculars $BE$ and $DF$ dropped from vertices $B$ and $D$ of parallelogram $ABCD$ onto sides $AD$ and $BC$, respectively, divide the parallelogram into three parts of equal area. A segment $DG$, equal to segment $BD$, is laid out on the extension of diagonal $BD$ beyond vertex $D$. Line $BE$ intersects segment $AG$ at point $H$. Find the ratio $AH: HG$. | 1:1 |
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agentica-org/DeepScaleR-Preview-Dataset | If the function $f(x)=\frac{1}{3}x^{3}-\frac{3}{2}x^{2}+ax+4$ is strictly decreasing on the interval $[-1,4]$, then the value of the real number $a$ is ______. | -4 |
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agentica-org/DeepScaleR-Preview-Dataset | Carolyn and Paul are playing a game starting with a list of the integers $1$ to $n.$ The rules of the game are:
$\bullet$ Carolyn always has the first turn.
$\bullet$ Carolyn and Paul alternate turns.
$\bullet$ On each of her turns, Carolyn must remove one number from the list such that this number has at least one positive divisor other than itself remaining in the list.
$\bullet$ On each of his turns, Paul must remove from the list all of the positive divisors of the number that Carolyn has just removed, and he can optionally choose one additional number that is a multiple of any divisor he is removing.
$\bullet$ If Carolyn cannot remove any more numbers, then Paul removes the rest of the numbers.
For example, if $n=8,$ a possible sequence of moves could be considered.
Suppose that $n=8$ and Carolyn removes the integer $4$ on her first turn. Determine the sum of the numbers that Carolyn removes. | 12 |
|
agentica-org/DeepScaleR-Preview-Dataset | A certain school randomly selected several students to investigate the daily physical exercise time of students in the school. They obtained data on the daily physical exercise time (unit: minutes) and organized and described the data. Some information is as follows:
- $a$. Distribution of daily physical exercise time:
| Daily Exercise Time $x$ (minutes) | Frequency (people) | Percentage |
|-----------------------------------|--------------------|------------|
| $60\leqslant x \lt 70$ | $14$ | $14\%$ |
| $70\leqslant x \lt 80$ | $40$ | $m$ |
| $80\leqslant x \lt 90$ | $35$ | $35\%$ |
| $x\geqslant 90$ | $n$ | $11\%$ |
- $b$. The daily physical exercise time in the group $80\leqslant x \lt 90$ is: $80$, $81$, $81$, $81$, $82$, $82$, $83$, $83$, $84$, $84$, $84$, $84$, $84$, $85$, $85$, $85$, $85$, $85$, $85$, $85$, $85$, $86$, $87$, $87$, $87$, $87$, $87$, $88$, $88$, $88$, $89$, $89$, $89$, $89$, $89$.
Based on the above information, answer the following questions:
$(1)$ In the table, $m=$______, $n=$______.
$(2)$ If the school has a total of $1000$ students, estimate the number of students in the school who exercise for at least $80$ minutes per day.
$(3)$ The school is planning to set a time standard $p$ (unit: minutes) to commend students who exercise for at least $p$ minutes per day. If $25\%$ of the students are to be commended, what value can $p$ be? | 86 |
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agentica-org/DeepScaleR-Preview-Dataset | Let's consider the number 2023. If 2023 were expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers? | 48 |
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agentica-org/DeepScaleR-Preview-Dataset | Find the numbers $\mathbf{1 5 3 , 3 7 0 , 3 7 1 , 4 0 7}$. | 153, 370, 371, 407 |
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agentica-org/DeepScaleR-Preview-Dataset | Santa Claus arrived at the house of Arnaldo and Bernaldo carrying ten distinct toys numbered from 1 to 10 and said to them: "Toy number 1 is for you, Arnaldo, and toy number 2 is for you, Bernaldo. But this year, you may choose to keep more toys as long as you leave at least one for me." Determine in how many ways Arnaldo and Bernaldo can distribute the remaining toys among themselves. | 6305 |
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agentica-org/DeepScaleR-Preview-Dataset | Given a matrix $\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22}\end{pmatrix}$ satisfies: $a_{11}$, $a_{12}$, $a_{21}$, $a_{22} \in \{0,1\}$, and $\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22}\end{vmatrix} =0$, determine the total number of distinct matrices. | 10 |
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agentica-org/DeepScaleR-Preview-Dataset | What is the sum of all the four-digit positive integers that end in 0? | 4945500 |
|
agentica-org/DeepScaleR-Preview-Dataset | Calculate $f(x) = 3x^5 + 5x^4 + 6x^3 - 8x^2 + 35x + 12$ using the Horner's Rule when $x = -2$. Find the value of $v_4$. | 83 |
|
agentica-org/DeepScaleR-Preview-Dataset | A set \( \mathcal{T} \) of distinct positive integers has the following property: for every integer \( y \) in \( \mathcal{T}, \) the arithmetic mean of the set of values obtained by deleting \( y \) from \( \mathcal{T} \) is an integer. Given that 2 belongs to \( \mathcal{T} \) and that 1024 is the largest element of \( \mathcal{T}, \) what is the greatest number of elements that \( \mathcal{T} \) can have? | 15 |
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agentica-org/DeepScaleR-Preview-Dataset | How many ordered pairs $(a,b)$ such that $a$ is a positive real number and $b$ is an integer between $2$ and $200$, inclusive, satisfy the equation $(\log_b a)^{2017}=\log_b(a^{2017})?$ | 597 |
|
agentica-org/DeepScaleR-Preview-Dataset |
The circle, which has its center on the hypotenuse $AB$ of the right triangle $ABC$, touches the two legs $AC$ and $BC$ at points $E$ and $D$ respectively.
Find the angle $ABC$, given that $AE = 1$ and $BD = 3$. | 30 |
|
agentica-org/DeepScaleR-Preview-Dataset | What is the smallest positive integer that is divisible by 111 and has the last four digits as 2004? | 662004 |
|
agentica-org/DeepScaleR-Preview-Dataset | An ellipse with equation
\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]contains the circles $(x - 1)^2 + y^2 = 1$ and $(x + 1)^2 +y^2 = 1.$ Then the smallest possible area of the ellipse can be expressed in the form $k \pi.$ Find $k.$ | \frac{3 \sqrt{3}}{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $A B C$ be a triangle with $A B=5, B C=8, C A=11$. The incircle $\omega$ and $A$-excircle $^{1} \Gamma$ are centered at $I_{1}$ and $I_{2}$, respectively, and are tangent to $B C$ at $D_{1}$ and $D_{2}$, respectively. Find the ratio of the area of $\triangle A I_{1} D_{1}$ to the area of $\triangle A I_{2} D_{2}$. | \frac{1}{9} |
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agentica-org/DeepScaleR-Preview-Dataset | The clock shows $00:00$, with both the hour and minute hands coinciding. Considering this coincidence as number 0, determine after what time interval (in minutes) they will coincide for the 21st time. If the answer is not an integer, round the result to the nearest hundredth. | 1374.55 |
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agentica-org/DeepScaleR-Preview-Dataset | Ten positive integers include the numbers 3, 5, 8, 9, and 11. What is the largest possible value of the median of this list of ten positive integers? | 11 |
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agentica-org/DeepScaleR-Preview-Dataset | Given a circle with center \(O\) and radius \(OD\) perpendicular to chord \(AB\), intersecting \(AB\) at point \(C\). Line segment \(AO\) is extended to intersect the circle at point \(E\). If \(AB = 8\) and \(CD = 2\), what is the area of \(\triangle BCE\)? | 12 |
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agentica-org/DeepScaleR-Preview-Dataset | Find the number of eight-digit numbers whose product of digits equals 1400. The answer must be presented as an integer. | 5880 |
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agentica-org/DeepScaleR-Preview-Dataset | In the diagram, the rectangle has a width $w$, a length of $8$, and a perimeter of $24$. What is the ratio of its width to its length? [asy]
pair a = (0, 0); pair b = (8, 0); pair c = (8, 4); pair d = (0, 4);
draw(a--b--c--d--cycle);
label("$w$", midpoint(a--d), W); label("$8$", midpoint(c--d), N);
[/asy] Write your answer in the form $x:y$, where $x$ and $y$ are relatively prime positive integers. | 1 : 2 |
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agentica-org/DeepScaleR-Preview-Dataset | A line contains the points $(-1, 6)$, $(6, k)$ and $(20, 3)$. What is the value of $k$? | 5 |
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agentica-org/DeepScaleR-Preview-Dataset | Find the smallest prime \( p > 100 \) for which there exists an integer \( a > 1 \) such that \( p \) divides \( \frac{a^{89} - 1}{a - 1} \). | 179 |
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agentica-org/DeepScaleR-Preview-Dataset | Find the largest six-digit number in which all digits are distinct, and each digit, except the first and last ones, is either the sum or the difference of its neighboring digits. | 972538 |
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agentica-org/DeepScaleR-Preview-Dataset | In a rectangular array of points, with 5 rows and $N$ columns, the points are numbered consecutively from left to right beginning with the top row. Thus the top row is numbered 1 through $N,$ the second row is numbered $N + 1$ through $2N,$ and so forth. Five points, $P_1, P_2, P_3, P_4,$ and $P_5,$ are selected so that each $P_i$ is in row $i.$ Let $x_i$ be the number associated with $P_i.$ Now renumber the array consecutively from top to bottom, beginning with the first column. Let $y_i$ be the number associated with $P_i$ after the renumbering. It is found that $x_1 = y_2,$ $x_2 = y_1,$ $x_3 = y_4,$ $x_4 = y_5,$ and $x_5 = y_3.$ Find the smallest possible value of $N.$
| 149 |
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agentica-org/DeepScaleR-Preview-Dataset | If $a>0$ and $b>0,$ a new operation $\nabla$ is defined as follows: $$a \nabla b = \frac{a + b}{1 + ab}.$$For example, $$3 \nabla 6 = \frac{3 + 6}{1 + 3 \times 6} = \frac{9}{19}.$$Calculate $(1 \nabla 2) \nabla 3.$ | 1 |
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agentica-org/DeepScaleR-Preview-Dataset | Two lines passing through point \( M \), which lies outside the circle with center \( O \), touch the circle at points \( A \) and \( B \). Segment \( OM \) is divided in half by the circle. In what ratio is segment \( OM \) divided by line \( AB \)? | 1:3 |
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agentica-org/DeepScaleR-Preview-Dataset | The height of a trapezoid, whose diagonals are mutually perpendicular, is 4. Find the area of the trapezoid if one of its diagonals is 5. | \frac{50}{3} |
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agentica-org/DeepScaleR-Preview-Dataset | Two numbers are such that their difference, their sum, and their product are to one another as $1:7:24$. The product of the two numbers is: | 48 |
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agentica-org/DeepScaleR-Preview-Dataset | I have 6 shirts, 6 pairs of pants, and 6 hats. Each item comes in the same 6 colors (so that I have one of each item of each color). I refuse to wear an outfit in which all 3 items are the same color. How many choices for outfits do I have? | 210 |
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agentica-org/DeepScaleR-Preview-Dataset | In triangle $\triangle ABC$, a line passing through the midpoint $E$ of the median $AD$ intersects sides $AB$ and $AC$ at points $M$ and $N$ respectively. Let $\overrightarrow{AM} = x\overrightarrow{AB}$ and $\overrightarrow{AN} = y\overrightarrow{AC}$ ($x, y \neq 0$), then the minimum value of $4x+y$ is \_\_\_\_\_\_. | \frac{9}{4} |
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agentica-org/DeepScaleR-Preview-Dataset | A 7' × 11' table sits in the corner of a square room. The table is to be rotated so that the side formerly 7' now lies along what was previously the end side of the longer dimension. Determine the smallest integer value of the side S of the room needed to accommodate this move. | 14 |
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agentica-org/DeepScaleR-Preview-Dataset | 24×12, my approach is to first calculate \_\_\_\_\_\_, then calculate \_\_\_\_\_\_, and finally calculate \_\_\_\_\_\_. | 288 |
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agentica-org/DeepScaleR-Preview-Dataset | How many times does the digit 9 appear in the list of all integers from 1 to 1000? | 300 |
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agentica-org/DeepScaleR-Preview-Dataset | Let \[f(x) =
\begin{cases}
9x+4 &\text{if }x\text{ is an integer}, \\
\lfloor{x}\rfloor+5 &\text{if }x\text{ is not an integer}.
\end{cases}
\]Find $f(\sqrt{29})$. | 10 |
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agentica-org/DeepScaleR-Preview-Dataset | A polynomial with integer coefficients is of the form
\[x^3 + a_2 x^2 + a_1 x - 11 = 0.\]Enter all the possible integer roots of this polynomial, separated by commas. | -11, -1, 1, 11 |
|
agentica-org/DeepScaleR-Preview-Dataset | Calculate the roundness of 1,728,000. | 19 |
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agentica-org/DeepScaleR-Preview-Dataset | Compute the square of 1017 without a calculator. | 1034289 |
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agentica-org/DeepScaleR-Preview-Dataset | There are five students: A, B, C, D, and E.
1. In how many different ways can they line up in a row such that A and B must be adjacent, and C and D cannot be adjacent?
2. In how many different ways can these five students be distributed into three classes, with each class having at least one student? | 150 |
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agentica-org/DeepScaleR-Preview-Dataset | Jo, Blair, and Parker take turns counting from 1, increasing by one more than the last number said by the previous person. What is the $100^{\text{th}}$ number said? | 100 |
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agentica-org/DeepScaleR-Preview-Dataset | Given a quadratic function in terms of \\(x\\), \\(f(x)=ax^{2}-4bx+1\\).
\\((1)\\) Let set \\(P=\\{1,2,3\\}\\) and \\(Q=\\{-1,1,2,3,4\\}\\), randomly pick a number from set \\(P\\) as \\(a\\) and from set \\(Q\\) as \\(b\\), calculate the probability that the function \\(y=f(x)\\) is increasing in the interval \\([1,+∞)\\).
\\((2)\\) Suppose point \\((a,b)\\) is a random point within the region defined by \\( \\begin{cases} x+y-8\\leqslant 0 \\\\ x > 0 \\\\ y > 0\\end{cases}\\), denote \\(A=\\{y=f(x)\\) has two zeros, one greater than \\(1\\) and the other less than \\(1\\}\\), calculate the probability of event \\(A\\) occurring. | \dfrac{961}{1280} |
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agentica-org/DeepScaleR-Preview-Dataset | Upon cutting a certain rectangle in half, you obtain two rectangles that are scaled down versions of the original. What is the ratio of the longer side length to the shorter side length? | \sqrt{2} |
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agentica-org/DeepScaleR-Preview-Dataset | Let $T_i$ be the set of all integers $n$ such that $50i \leq n < 50(i + 1)$. For example, $T_4$ is the set $\{200, 201, 202, \ldots, 249\}$. How many of the sets $T_0, T_1, T_2, \ldots, T_{1999}$ do not contain a perfect square? | 1733 |
|
agentica-org/DeepScaleR-Preview-Dataset | The fare in Moscow with the "Troika" card in 2016 is 32 rubles for one trip on the metro and 31 rubles for one trip on ground transportation. What is the minimum total number of trips that can be made at these rates, spending exactly 5000 rubles? | 157 |
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agentica-org/DeepScaleR-Preview-Dataset | In three-dimensional space, find the number of lattice points that have a distance of 5 from the origin.
Note: A point is a lattice point if all its coordinates are integers. | 54 |
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agentica-org/DeepScaleR-Preview-Dataset | What is the coefficient of $x^3y^5$ in the expansion of $\left(\frac{2}{3}x - \frac{y}{3}\right)^8$? | -\frac{448}{6561} |
|
agentica-org/DeepScaleR-Preview-Dataset | In rectangle $ABCD$, angle $C$ is trisected by $\overline{CF}$ and $\overline{CE}$, where $E$ is on $\overline{AB}$, $F$ is on $\overline{AD}$, $BE=6$, and $AF=2$. Find the area of $ABCD$.
[asy]
import olympiad; import geometry; size(150); defaultpen(linewidth(0.8)); dotfactor=4;
real length = 2 * (6*sqrt(3) - 2), width = 6*sqrt(3);
draw(origin--(length,0)--(length,width)--(0,width)--cycle);
draw((length,width)--(0,2)^^(length,width)--(length - 6,0));
dot("$A$",origin,SW); dot("$B$",(length,0),SE); dot("$C$",(length,width),NE); dot("$D$",(0,width),NW); dot("$F$",(0,2),W); dot("$E$",(length - 6,0),S);
[/asy] | 108\sqrt{3}-36 |
|
agentica-org/DeepScaleR-Preview-Dataset | Point $P$ is on the ellipse $\frac{x^{2}}{16}+ \frac{y^{2}}{9}=1$. The maximum and minimum distances from point $P$ to the line $3x-4y=24$ are $\_\_\_\_\_\_$. | \frac{12(2- \sqrt{2})}{5} |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $a$ and $b$ be positive real numbers such that $a + 2b = 1.$ Find the minimum value of
\[\frac{1}{a} + \frac{1}{b}.\] | 3 + 2 \sqrt{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | On a computer screen is the single character a. The computer has two keys: c (copy) and p (paste), which may be pressed in any sequence. Pressing p increases the number of a's on screen by the number that were there the last time c was pressed. c doesn't change the number of a's on screen. Determine the fewest number of keystrokes required to attain at least 2018 a's on screen. (Note: pressing p before the first press of c does nothing). | 21 |
|
agentica-org/DeepScaleR-Preview-Dataset | Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area? | \frac{223}{286} |
|
agentica-org/DeepScaleR-Preview-Dataset | If $a+\frac {a} {3}=\frac {8} {3}$, what is the value of $a$? | 2 |
|
agentica-org/DeepScaleR-Preview-Dataset | A clock takes $7$ seconds to strike $9$ o'clock starting precisely from $9:00$ o'clock. If the interval between each strike increases by $0.2$ seconds as time progresses, calculate the time it takes to strike $12$ o'clock. | 12.925 |
|
agentica-org/DeepScaleR-Preview-Dataset | A rectangle can be divided into \( n \) equal squares. The same rectangle can also be divided into \( n + 76 \) equal squares. Find all possible values of \( n \). | 324 |
|
agentica-org/DeepScaleR-Preview-Dataset | A total of $731$ objects are put into $n$ nonempty bags where $n$ is a positive integer. These bags can be distributed into $17$ red boxes and also into $43$ blue boxes so that each red and each blue box contain $43$ and $17$ objects, respectively. Find the minimum value of $n$ . | 17 |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $ABCD$ be a parallelogram with area $15$. Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC.$ See the figure, which also shows the relative locations of these points.
Suppose $PQ=6$ and $RS=8,$ and let $d$ denote the length of $\overline{BD},$ the longer diagonal of $ABCD.$ Then $d^2$ can be written in the form $m+n\sqrt p,$ where $m,n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m+n+p?$ | 81 |
|
agentica-org/DeepScaleR-Preview-Dataset | If $3x+7\equiv 2\pmod{16}$, then $2x+11$ is congruent $\pmod{16}$ to what integer between $0$ and $15$, inclusive? | 13 |
|
agentica-org/DeepScaleR-Preview-Dataset | Let's define a number as complex if it has at least two different prime divisors. Find the greatest natural number that cannot be represented as the sum of two complex numbers. | 23 |
|
agentica-org/DeepScaleR-Preview-Dataset | The product of positive integers $x$, $y$ and $z$ equals 2004. What is the minimum possible value of the sum $x + y + z$? | 174 |
|
agentica-org/DeepScaleR-Preview-Dataset | Find the number of integer points that satisfy the system of inequalities:
\[
\begin{cases}
y \leqslant 3x \\
y \geqslant \frac{1}{3}x \\
x + y \leqslant 100
\end{cases}
\] | 2551 |
|
agentica-org/DeepScaleR-Preview-Dataset | The area of a square inscribed in a semicircle compared to the area of a square inscribed in a full circle. | 2:5 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given $\{a_{n}\}$ is a geometric sequence, $a_{2}a_{4}a_{5}=a_{3}a_{6}$, $a_{9}a_{10}=-8$, then $a_{7}=\_\_\_\_\_\_$. | -2 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given a polynomial \( P(x) \) with integer coefficients. It is known that \( P(1) = 2013 \), \( P(2013) = 1 \), and \( P(k) = k \), where \( k \) is some integer. Find \( k \). | 1007 |
|
agentica-org/DeepScaleR-Preview-Dataset | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $\cos \frac{A}{2}= \frac{2 \sqrt{5}}{5}, \overrightarrow{AB} \cdot \overrightarrow{AC}=15$.
$(1)$ Find the area of $\triangle ABC$;
$(2)$ If $\tan B=2$, find the value of $a$. | 2 \sqrt{5} |
|
agentica-org/DeepScaleR-Preview-Dataset | Find the integer $n$ such that $-150 < n < 150$ and $\tan n^\circ = \tan 286^\circ$. | -74 |
|
agentica-org/DeepScaleR-Preview-Dataset | Evaluate $81^{-\frac{1}{4}} + 16^{-\frac{3}{4}}$. Express your answer as a common fraction. | \frac{11}{24} |
|
agentica-org/DeepScaleR-Preview-Dataset | Given that two children, A and B, and three adults, 甲, 乙, and 丙, are standing in a line, A is not at either end, and exactly two of the three adults are standing next to each other. The number of different arrangements is $\boxed{\text{answer}}$. | 48 |
|
agentica-org/DeepScaleR-Preview-Dataset | In acute \\(\triangle ABC\\), the sides opposite to angles \\(A\\), \\(B\\), and \\(C\\) are \\(a\\), \\(b\\), and \\(c\\) respectively, with \\(a=4\\), \\(b=5\\), and the area of \\(\triangle ABC\\) is \\(5\sqrt{3}\\). Find the value of side \\(c=\\) ______. | \sqrt{21} |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $[r,s]$ denote the least common multiple of positive integers $r$ and $s$. Find the number of ordered triples $(a,b,c)$ of positive integers for which $[a,b] = 1000$, $[b,c] = 2000$, and $[c,a] = 2000$.
| 70 |
|
agentica-org/DeepScaleR-Preview-Dataset | Suppose that $f(x+3)=3x^2 + 7x + 4$ and $f(x)=ax^2 + bx + c$. What is $a+b+c$? | 2 |
|
agentica-org/DeepScaleR-Preview-Dataset | The equation $a^7xy-a^6y-a^5x=a^4(b^4-1)$ is equivalent to the equation $(a^mx-a^n)(a^py-a^2)=a^4b^4$ for some integers $m$, $n$, and $p$. Find $mnp$. | 24 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given that the polynomial \(x^2 - kx + 24\) has only positive integer roots, find the average of all distinct possibilities for \(k\). | 15 |
|
agentica-org/DeepScaleR-Preview-Dataset | If $AAA_4$ can be expressed as $33_b$, where $A$ is a digit in base 4 and $b$ is a base greater than 5, what is the smallest possible sum $A+b$? | 7 |
|
agentica-org/DeepScaleR-Preview-Dataset | Given an increasing geometric sequence $\{a_{n}\}$ with a common ratio greater than $1$ such that $a_{2}+a_{4}=20$, $a_{3}=8$.<br/>$(1)$ Find the general formula for $\{a_{n}\}$;<br/>$(2)$ Let $b_{m}$ be the number of terms of $\{a_{n}\}$ in the interval $\left(0,m\right]\left(m\in N*\right)$. Find the sum of the first $100$ terms of the sequence $\{b_{m}\}$, denoted as $S_{100}$. | 480 |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $x$ be a real number. Find the maximum value of $2^{x(1-x)}$. | \sqrt[4]{2} |
|
agentica-org/DeepScaleR-Preview-Dataset | The highest power of 2 that is a factor of \(15.13^{4} - 11^{4}\) needs to be determined. | 32 |
|
agentica-org/DeepScaleR-Preview-Dataset | David and Evan are playing a game. Evan thinks of a positive integer $N$ between 1 and 59, inclusive, and David tries to guess it. Each time David makes a guess, Evan will tell him whether the guess is greater than, equal to, or less than $N$. David wants to devise a strategy that will guarantee that he knows $N$ in five guesses. In David's strategy, each guess will be determined only by Evan's responses to any previous guesses (the first guess will always be the same), and David will only guess a number which satisfies each of Evan's responses. How many such strategies are there? | 36440 |
|
agentica-org/DeepScaleR-Preview-Dataset | How many integers $m$ are there from 1 to 1996, such that $\frac{m^{2}+7}{m+4}$ is not a reduced fraction? | 86 |
|
agentica-org/DeepScaleR-Preview-Dataset | For any real number $t$ , let $\lfloor t \rfloor$ denote the largest integer $\le t$ . Suppose that $N$ is the greatest integer such that $$ \left \lfloor \sqrt{\left \lfloor \sqrt{\left \lfloor \sqrt{N} \right \rfloor}\right \rfloor}\right \rfloor = 4 $$ Find the sum of digits of $N$ . | 24 |
|
agentica-org/DeepScaleR-Preview-Dataset | Let $S$ be the set $\{1,2,3,...,19\}$. For $a,b \in S$, define $a \succ b$ to mean that either $0 < a - b \le 9$ or $b - a > 9$. How many ordered triples $(x,y,z)$ of elements of $S$ have the property that $x \succ y$, $y \succ z$, and $z \succ x$? | 855 |
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agentica-org/DeepScaleR-Preview-Dataset | Calculate the following product:
$$\frac{1}{3}\times9\times\frac{1}{27}\times81\times\frac{1}{243}\times729\times\frac{1}{2187}\times6561\times\frac{1}{19683}\times59049.$$ | 243 |
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agentica-org/DeepScaleR-Preview-Dataset | Define the *hotel elevator cubic*as the unique cubic polynomial $P$ for which $P(11) = 11$ , $P(12) = 12$ , $P(13) = 14$ , $P(14) = 15$ . What is $P(15)$ ?
*Proposed by Evan Chen* | 13 |
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agentica-org/DeepScaleR-Preview-Dataset | In a right triangle $PQR$ where $\angle R = 90^\circ$, the lengths of sides $PQ = 15$ and $PR = 9$. Find $\sin Q$ and $\cos Q$. | \frac{3}{5} |
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agentica-org/DeepScaleR-Preview-Dataset | How many positive integers at most 420 leave different remainders when divided by each of 5, 6, and 7? | 250 |
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agentica-org/DeepScaleR-Preview-Dataset | The number of students in Carlos' graduating class is more than 100 and fewer than 200 and is 2 less than a multiple of 4, 3 less than a multiple of 5, and 4 less than a multiple of 6. How many students are in Carlos' graduating class? | 182 |
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agentica-org/DeepScaleR-Preview-Dataset | Solve for $y$: $4+2.3y = 1.7y - 20$ | -40 |
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agentica-org/DeepScaleR-Preview-Dataset | When \( N \) takes all the values from 1, 2, 3, \ldots, 2015, how many numbers of the form \( 3^{n} + n^{3} \) are divisible by 7? | 288 |
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agentica-org/DeepScaleR-Preview-Dataset | a) Vanya flips a coin 3 times, and Tanya flips a coin 2 times. What is the probability that Vanya gets more heads than Tanya?
b) Vanya flips a coin $n+1$ times, and Tanya flips a coin $n$ times. What is the probability that Vanya gets more heads than Tanya? | \frac{1}{2} |
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agentica-org/DeepScaleR-Preview-Dataset | Given the function $f(x)=e^{ax}-x-1$, where $a\neq 0$. If $f(x)\geqslant 0$ holds true for all $x\in R$, then the set of possible values for $a$ is \_\_\_\_\_\_. | \{1\} |
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agentica-org/DeepScaleR-Preview-Dataset | Given a sector with a central angle of $\alpha$ and a radius of $r$.
$(1)$ If $\alpha = 60^{\circ}$ and $r = 3$, find the arc length of the sector.
$(2)$ If the perimeter of the sector is $16$, at what angle $\alpha$ will the area of the sector be maximized? Also, find the maximum area. | 16 |
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agentica-org/DeepScaleR-Preview-Dataset | A right circular cone with a base radius $r$ and height $h$ lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base connects with the table traces a circular arc centered at the vertex of the cone. The cone first returns to its original position after making $20$ complete rotations. The value of $h/r$ in simplest form can be expressed as $\lambda\sqrt{k}$, where $\lambda$ and $k$ are positive integers, and $k$ is not divisible by the square of any prime. Find $\lambda + k$. | 400 |
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agentica-org/DeepScaleR-Preview-Dataset | Given a set of points in space, a *jump* consists of taking two points, $P$ and $Q,$ and replacing $P$ with the reflection of $P$ over $Q$ . Find the smallest number $n$ such that for any set of $n$ lattice points in $10$ -dimensional-space, it is possible to perform a finite number of jumps so that some two points coincide.
*Author: Anderson Wang* | 1025 |
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agentica-org/DeepScaleR-Preview-Dataset | Define an ordered triple $(A, B, C)$ of sets to be minimally intersecting if $|A \cap B| = |B \cap C| = |C \cap A| = 1$ and $A \cap B \cap C = \emptyset$. For example, $(\{1,2\},\{2,3\},\{1,3,4\})$ is a minimally intersecting triple. Let $N$ be the number of minimally intersecting ordered triples of sets for which each set is a subset of $\{1,2,3,4,5,6,7\}$. Find the remainder when $N$ is divided by $1000$.
| 760 |
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agentica-org/DeepScaleR-Preview-Dataset | Given the following system of equations: $$ \begin{cases} R I +G +SP = 50 R I +T + M = 63 G +T +SP = 25 SP + M = 13 M +R I = 48 N = 1 \end{cases} $$
Find the value of L that makes $LMT +SPR I NG = 2023$ true.
| \frac{341}{40} |
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agentica-org/DeepScaleR-Preview-Dataset | The number \(abcde\) has five distinct digits, each different from zero. When this number is multiplied by 4, the result is a five-digit number \(edcba\), which is the reverse of \(abcde\). What is the value of \(a + b + c + d + e\)? | 27 |
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agentica-org/DeepScaleR-Preview-Dataset | The villages Arkadino, Borisovo, and Vadimovo are connected by straight roads in pairs. Adjacent to the road between Arkadino and Borisovo is a square field, one side of which completely coincides with this road. Adjacent to the road between Borisovo and Vadimovo is a rectangular field, one side of which completely coincides with this road, and the other side is 4 times longer. Adjacent to the road between Arkadino and Vadimovo is a rectangular forest, one side of which completely coincides with this road, and the other side is 12 km. The area of the forest is 45 square km more than the sum of the areas of the fields. Find the total area of the forest and fields in square km. | 135 |
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